Manuscript Psy02-A@1 · Psy02-A.tex · proposition · d92ce402

Proposition: Concentration and coherence have different trajectories

Environmental dynamics and the two-budget principle › An explicit attenuator and its separate budgets · lines 1157–1193 · proof included or linked

Psy02-A@1#prop:attenuator-witness

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For the preceding input,

n (A_{η} (ρ_{+})) = \frac{η}{2}, D (ρ_{+}; A_{η}, Δ) = \frac{η}{2} .

With $η (s) = e^{- κ s}$ , $κ > 0$ , the selected observable concentration bound is

ϵ (2 e^{- κ s} + 2) \norm v^{2},

whereas the witness is

D_{s} = \frac{1}{2} e^{- κ s /2} .

For $0 < d_{m i n} \leq 1/2$ , witness survival up to $T$ is equivalent to

T \leq \frac{2}{κ} lo g \frac{1}{2 d _{m i n}} .

Proof. Vacuum attenuation maps

∣ 0 ⟩ ⟨ 1 ∣ ⟼ η ∣ 0 ⟩ ⟨ 1 ∣, ∣ 1 ⟩ ⟨ 1 ∣ ⟼ η ∣ 1 ⟩ ⟨ 1 ∣ + (1 - η) ∣ 0 ⟩ ⟨ 0 ∣ .

The output difference between the coherent and dephased inputs has off-diagonal entries $η /2$ , hence eigenvalues $\pm η /2$ . The remaining statements follow by substitution.

Displayed formulas in this passage

unlabeled · line 1160
$n (A_{η} (ρ_{+})) = \frac{η}{2}, D (ρ_{+}; A_{η}, Δ) = \frac{η}{2} .$
unlabeled · line 1167
$ϵ (2 e^{- κ s} + 2) \norm v^{2},$
unlabeled · line 1171
$D_{s} = \frac{1}{2} e^{- κ s /2} .$
unlabeled · line 1176
$T \leq \frac{2}{κ} lo g \frac{1}{2 d _{m i n}} .$
unlabeled · line 1183
$∣ 0 ⟩ ⟨ 1 ∣ ⟼ η ∣ 0 ⟩ ⟨ 1 ∣, ∣ 1 ⟩ ⟨ 1 ∣ ⟼ η ∣ 1 ⟩ ⟨ 1 ∣ + (1 - η) ∣ 0 ⟩ ⟨ 0 ∣ .$

Proposition: Concentration and coherence have different trajectories

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